Not only did he grow up in a kisaeng house as the child of a kisaeng, his only skill is as a musician. He's also uneducated, small, and pretty, which makes him highly likely to be victimized. And his social class means that no one would care if he lived or died.
He can't just easily go out and get a job. He is already saving money the best he can. In the meantime, his boss owns his home, pays for his food, pays for his mom's home and care, and pays for her medicine. His only friends also work for the same guy. If he left, all of that would automatically go away and he and his mom would both be on the streets with almost nothing. She could die with no medicine and no one to watch over her while he tried to find work.
I agree he's both foolish and extremely naive as a character, but his options are also incredibly limited and that's what's making him so scared.
My thoughts here are: knowing how your lover clearly holds status and is willing to extend various offers of luxuries (even offering a HOUSE) he didn’t have enough foresight to recognize what leaving the pleasure house actually entailed to ask about being connected to a job through his lover.
Even before the master took him to see his mom, what was his plan? Because what you said is true, but he would have known all that and still planned to leave so..? How did he plan to live until that point. He was going to find a job on the outside and support himself surely? Or maybe play house with his lover?
Then my point stands that he’s not being rational. If he’d just be transparent with his lover about his sick mother and the additional expenses for him to be independent, he’d easily resolve which path to take. Obviously, we know ML wouldn’t let the MC work and would have the mother handled but that’s the POINT. The smart way to go about it is figuring out how to secure the support, yet he goes and gives away what foundation he already built up to that point without explaining anything to ML regarding the roots of his anxiety. Not to mention he’s just been hearing about (and then seeing the living example through his mom) how fleeting and fickle love can be and how it can ruin you.
I didn’t mean to ramble like this lol sorry but I do see your point

What’s the first step one takes in becoming independent while learning they may be taking on a sick relative? Perhaps securing additional jobs to prepare adequate income for household stability?
Nope! Going to your wealthy lover and giving him all your life savings. Like what a dumbass.